Rosh Chodesh is considered a minor holiday, and it is the first mitzvah given to the Israelites when they left Egypt:
1 And Hashem spoke unto Moses and Aaron in the land of Egypt, saying:That was the month of Nisan! And like the Israelites journeying out of Egypt in Nisan, so will we experience redemption and begin anew the journey to Har Sinai in the middle of this month!
2 "This month shall be unto you the beginning of months; it shall be the first month of the year to you. ..."
But back to Rosh Chodesh in general.
When there was a Sanhedrin, two kosher witnesses would report the new moon to the Sanhedrin (a beit din), who would then declare the new month to start on that day. Going back to the diaspora post, the new month used to be announced by bonfires, and sometimes there was doubt in those outlying communities as to on which day the month would begin. Therefore, the outlying communities observed the beginning of the month on both possible days it could be. Because of this, every month has either 29 or 30 days. There is a calculation for that, but it's not important to know, at least not for most of you. If you need to know for some reason, check your local Jewish calendar!
The new month is "announced" in synagogue at Shabbat morning services on the Shabbat before the appearance of the new moon. On the day(s) of Rosh Chodesh, special language is added to the daily prayers and the grace after meals (birkat hamazon). There is a Torah reading for Rosh Chodesh, and Mussaf is added after shacharis that morning.
Rosh Chodesh is linked to women, and today, many communities have special women-only celebrations for Rosh Chodesh. Generally, the women will gather at someone's home with food, drinks, and either have a discussion or a class. They may also recite Tehillim for the ill, shidduchs needed, or others who need some sort of heavenly assistance.